-# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
-# have to explicitly mangle our sitedir here so that "mjo.cone"
-# resolves.
-from os.path import abspath
-from site import addsitedir
-addsitedir(abspath('../../'))
-
from sage.all import *
def rearrangement_cone(p,n):
A polyhedral closed convex cone object representing a rearrangement
cone of order ``p`` in ``n`` dimensions.
+ REFERENCES:
+
+ .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
+ Inradius and Circumradius of Various Convex Cones Arising in
+ Applications. Set-Valued and Variational Analysis, 18(3-4),
+ 483-511, 2010. doi:10.1007/s11228-010-0150-z
+
+ .. [GowdaJeong] Muddappa Seetharama Gowda and Juyoung Jeong.
+ Spectral cones in Euclidean Jordan algebras.
+ Linear Algebra and its Applications, 509, 286-305.
+ doi:10.1016/j.laa.2016.08.004
+
+ .. [Jeong] Juyoung Jeong.
+ Spectral sets and functions on Euclidean Jordan algebras.
+
+ SETUP::
+
+ sage: from mjo.cone.rearrangement import rearrangement_cone
+
EXAMPLES:
The rearrangement cones of order one are nonnegative orthants::
sage: rearrangement_cone(5,5).lineality()
4
+ All rearrangement cones are proper::
+
+ sage: all( rearrangement_cone(p,n).is_proper()
+ ....: for n in xrange(10)
+ ....: for p in xrange(1, n) )
+ True
+
+ The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
+ dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise::
+
+ sage: all( rearrangement_cone(p,n).lyapunov_rank() == n
+ ....: for n in xrange(2, 10)
+ ....: for p in [1, n-1] )
+ True
+ sage: all( rearrangement_cone(p,n).lyapunov_rank() == 1
+ ....: for n in xrange(3, 10)
+ ....: for p in xrange(2, n-1) )
+ True
+
TESTS:
- todo.
- should be permutation invariant.
- should have the expected lyapunov rank.
- just loop through them all for n <= 10 and p < n?
+ The rearrangement cone is permutation-invariant::
+
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: p = ZZ.random_element(1,n)
+ sage: K = rearrangement_cone(p,n)
+ sage: P = SymmetricGroup(n).random_element().matrix()
+ sage: all( K.contains(P*r) for r in K )
+ True
+
+ The smallest ``p`` components of every element of the rearrangement
+ cone should sum to a nonnegative number (this tests that the
+ generators really are what we think they are)::
+
+ sage: set_random_seed()
+ sage: def _has_rearrangement_property(v,p):
+ ....: return sum( sorted(v)[0:p] ) >= 0
+ sage: all( _has_rearrangement_property(
+ ....: rearrangement_cone(p,n).random_element(),
+ ....: p
+ ....: )
+ ....: for n in xrange(2, 10)
+ ....: for p in xrange(1, n-1)
+ ....: )
+ True
+
+ The rearrangenent cone of order ``p`` is contained in the
+ rearrangement cone of order ``p + 1``::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10)
+ sage: p = ZZ.random_element(1,n)
+ sage: K1 = rearrangement_cone(p,n)
+ sage: K2 = rearrangement_cone(p+1,n)
+ sage: all( x in K2 for x in K1 )
+ True
+
+ The order ``p`` should be between ``1`` and ``n``, inclusive::
+
+ sage: rearrangement_cone(0,3)
+ Traceback (most recent call last):
+ ...
+ ValueError: order p=0 should be between 1 and n=3, inclusive
+ sage: rearrangement_cone(5,3)
+ Traceback (most recent call last):
+ ...
+ ValueError: order p=5 should be between 1 and n=3, inclusive
"""
+ if p < 1 or p > n:
+ raise ValueError('order p=%d should be between 1 and n=%d, inclusive'
+ %
+ (p,n))
def d(j):
v = [1]*n # Create the list of all ones...
v[j] = 1 - p # Now "fix" the ``j``th entry.
return v
- V = VectorSpace(QQ, n)
- G = V.basis() + [ d(j) for j in range(n) ]
+ G = identity_matrix(ZZ,n).rows() + [ d(j) for j in xrange(n) ]
return Cone(G)